In some applications, we may instead want to find the measurements that minimize the reconstruction Lipschitz constant \(\Vert \Phi _{S}\Vert _{\mathcal {X}_N, \text{lip}}\)  using a fixed sensor budget \(\#({S}) \le C\) .
By running the greedy algorithm repeatedly using different thresholds \(L\)  it is possible to obtain upper and sometimes lower bounds on this budget-constrained minimum Lipschitz constant \(L^*\) .
This idea is closely related to the approach of [1]}.
If the greedy algorithm using Lipschitz constant \(L\)  chooses sensors \({S}\)  that meet the budget \(\#({S}) \le C\)  then \(L\)  is obviously an upper bound on \(L^*\) .
In practice, we can use a bisection search over \(L\)  to find nearly the smallest \(L\)  to any given tolerance for which \(\#({S}) \le C\) .
To get the lower bound, the greedy algorithm is run with a small enough \(L\)  so that the bound on the minimum possible cost from Eq.  REF exceeds the budget
\(C < \#({S})/(1 + \ln \kappa ).\) 
